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Refined theorems of the Birch and Swinnerton-Dyer type

Ki-Seng Tan (1995)

Annales de l'institut Fourier

In this paper, we generalize the context of the Mazur-Tate conjecture and sharpen, in a certain way, the statement of the conjecture. Our main result will be to establish the truth of a part of these new sharpened conjectures, provided that one assume the truth of the classical Birch and Swinnerton-Dyer conjectures. This is particularly striking in the function field case, where these results can be viewed as being a refinement of the earlier work of Tate and Milne.

Regular analytic transformations of 2

Joseph Gubeladze (2000)

Annales Polonici Mathematici

Existence of loops for non-injective regular analytic transformations of the real plane is shown. As an application, a criterion for injectivity of a regular analytic transformation of 2 in terms of the Jacobian and the first and second order partial derivatives is obtained. This criterion is new even in the special case of polynomial transformations.

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